Average Error: 33.2 → 10.0
Time: 43.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.896735429482593 \cdot 10^{+46}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.1815645801505244 \cdot 10^{-79}:\\ \;\;\;\;\left(\sqrt{\left(c \cdot -4\right) \cdot a + b \cdot b} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -4.896735429482593 \cdot 10^{+46}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 1.1815645801505244 \cdot 10^{-79}:\\
\;\;\;\;\left(\sqrt{\left(c \cdot -4\right) \cdot a + b \cdot b} - b\right) \cdot \frac{1}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;-\frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r14562282 = b;
        double r14562283 = -r14562282;
        double r14562284 = r14562282 * r14562282;
        double r14562285 = 4.0;
        double r14562286 = a;
        double r14562287 = r14562285 * r14562286;
        double r14562288 = c;
        double r14562289 = r14562287 * r14562288;
        double r14562290 = r14562284 - r14562289;
        double r14562291 = sqrt(r14562290);
        double r14562292 = r14562283 + r14562291;
        double r14562293 = 2.0;
        double r14562294 = r14562293 * r14562286;
        double r14562295 = r14562292 / r14562294;
        return r14562295;
}
↓
double f(double a, double b, double c) {
        double r14562296 = b;
        double r14562297 = -4.896735429482593e+46;
        bool r14562298 = r14562296 <= r14562297;
        double r14562299 = c;
        double r14562300 = r14562299 / r14562296;
        double r14562301 = a;
        double r14562302 = r14562296 / r14562301;
        double r14562303 = r14562300 - r14562302;
        double r14562304 = 1.1815645801505244e-79;
        bool r14562305 = r14562296 <= r14562304;
        double r14562306 = -4.0;
        double r14562307 = r14562299 * r14562306;
        double r14562308 = r14562307 * r14562301;
        double r14562309 = r14562296 * r14562296;
        double r14562310 = r14562308 + r14562309;
        double r14562311 = sqrt(r14562310);
        double r14562312 = r14562311 - r14562296;
        double r14562313 = 1.0;
        double r14562314 = 2.0;
        double r14562315 = r14562301 * r14562314;
        double r14562316 = r14562313 / r14562315;
        double r14562317 = r14562312 * r14562316;
        double r14562318 = -r14562300;
        double r14562319 = r14562305 ? r14562317 : r14562318;
        double r14562320 = r14562298 ? r14562303 : r14562319;
        return r14562320;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original33.2
Target20.4
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.896735429482593e+46

    1. Initial program 34.6

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified34.5

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}}\]

    3. Taylor expanded around 0 34.5

      \[\leadsto \frac{\sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}} - b}{2 \cdot a}\]

    4. Simplified34.6

      \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b + \left(c \cdot -4\right) \cdot a}} - b}{2 \cdot a}\]

    5. Taylor expanded around -inf 5.4

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]

    if -4.896735429482593e+46 < b < 1.1815645801505244e-79

    1. Initial program 12.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified12.8

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}}\]

    3. Taylor expanded around 0 12.8

      \[\leadsto \frac{\sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}} - b}{2 \cdot a}\]

    4. Simplified12.8

      \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b + \left(c \cdot -4\right) \cdot a}} - b}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied div-inv12.9

      \[\leadsto \color{blue}{\left(\sqrt{b \cdot b + \left(c \cdot -4\right) \cdot a} - b\right) \cdot \frac{1}{2 \cdot a}}\]

    if 1.1815645801505244e-79 < b

    1. Initial program 52.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified52.3

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}}\]

    3. Taylor expanded around inf 9.6

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

    4. Simplified9.6

      \[\leadsto \color{blue}{-\frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.896735429482593 \cdot 10^{+46}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.1815645801505244 \cdot 10^{-79}:\\ \;\;\;\;\left(\sqrt{\left(c \cdot -4\right) \cdot a + b \cdot b} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))